On Generalized Signal Waveforms for Satellite Navigation (797942), страница 26
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Putting now (4.78) and (4.80) together, the evolution of the phase over oneperiod n , that is nTc ≤ t ≤ (n + 1)T , will be:φn (t ) = π ∑ ckkwhich can also be expressed as follows:φn (t ) = π ⋅ cn ⋅t∫ f (τ − nT )dτ(4.81)t − nTc π n −1+ ∑ ck2Tc2 k = −∞(4.82)c−∞Accordingly, the expression of the MSK will be:⎧⎪ ⎡⎧⎪ ⎡⎛ t − nTc ⎞ π n −1 ⎤ ⎫⎪⎛ t − nTc ⎞⎤ ⎫⎪⎟⎟ + ∑ ck ⎥ ⎬ = A ⋅ exp⎨ j ⎢θ n −1 + π ⋅ cn ⎜⎜⎟⎟⎥ ⎬ (4.83)sn (t ) = A ⋅ exp⎨ j ⎢π ⋅ cn ⎜⎜⎪⎩ ⎣⎪⎩ ⎣⎝ 2Tc ⎠ 2 k = −∞ ⎦ ⎪⎭⎝ 2Tc ⎠⎦ ⎪⎭with θ n −1 =πn −1∑c2 k = −∞k.104GNSS Signal StructureTaking now the real and imaginary part of (4.83), the sinusoidal shape of the chip waveformis then observed.
Since we assume ideal codes, we only need to work with the chip waveformas described in (4.8). Next figure compares the typical binary pulse with that of MSK:Figure 4.21. Binary and MSK chip waveformIn order the original power of MSK to remain equivalent to that of the ordinary binarysignals, the factor A = 2 was used in the MSK expression as we can observe. To derivenow the PSD according to (4.8), we calculate first the Fourier Transform of the MSK pulsewaveform:πtTscTsc−j⎤⎡ j TπtTsc22sc⎛ πt ⎞ -jωt+e⎥ -jωt⎢e⎟⎟e dt = 2 ∫ ⎢(4.84)S MSK (ω ) = ∫ 2 cos⎜⎜⎥e dtTsc ⎠2TscTsc⎝−−⎥22 ⎢⎦⎣which simplifies to:⎧ ⎡⎛ π⎡⎛ π⎞T ⎤⎞T ⎤⎫− ω ⎟⎟ sc ⎥ sin ⎢⎜⎜+ ω ⎟⎟ sc ⎥ ⎪⎪ sin ⎢⎜⎜⎪⎝ Tsc⎠ 2 ⎦⎠ 2 ⎦⎪⎣⎝ Tsc+S MSK (ω ) = j 2 ⎨ ⎣(4.85)⎬⎛⎞⎛⎞ππ⎪⎪⎜⎜⎜⎜− ω ⎟⎟+ ω ⎟⎟⎪⎪TT⎝ sc⎠⎝ sc⎠⎩⎭As we can recognize, we are defining the spreading waveform in a subchip of length Tscwhere Tc = nTsc , being Tc the duration of a chip and n the number of subchips in one chip.Accordingly, this is the Fourier transform of the subchip part and f sc = nf c .
This expressioncan be further simplified yielding:⎧⎫⎛ ωT ⎞2 2π cos⎜ sc ⎟⎪⎪11⎛ ωT ⎞ ⎪⎪⎝ 2 ⎠S MSK (ω ) = j 2 cos⎜ sc ⎟⎨+⎬= j2⎞⎛π⎞⎞ ⎛π⎝ 2 ⎠⎪ ⎛ π⎜⎜+ ω ⎟⎟ ⎪− ω ⎟⎟ ⎜⎜Tsc ⎜⎜ 2 − ω 2 ⎟⎟⎪ ⎝ Tsc⎠ ⎪⎭⎠ ⎝ Tsc⎝ Tsc⎠⎩(4.86)According to this, the normalized Power Spectral Density of the spreading MSK waveformadopts the following form:105GNSS Signal StructureGMSK (ω ) =12S MSK (ω )Tsc⎛ ωT ⎞8π 2 cos 2 ⎜ sc ⎟1⎝ 2 ⎠=22Tsc 2 ⎛ π2⎞Tsc ⎜⎜ 2 − ω ⎟⎟⎝ Tsc⎠(4.87)which can also be expressed as follows:G MSK ( f ) =8 f sc3π2(⎛ πf ⎞⎟⎟cos 2 ⎜⎜f⎝ sc ⎠22f sc − 4 f 2(4.88))As we can recognize, this expression perfectly coincides with that derived in[S. Pasupathy, 1979].Once we have derived the pulse waveform for a cosine shape, any SMCS can be obtainedusing the general formula obtained in (4.28). As an example, in the next lines we derive theexpression for a sine-phased BOC(fs , fc) with MSK pulses and we will compare it with theoriginal BPSK with MSK pulses.
For simplicity in the notation we call MSK(fs , fc) to the firstone and MSK(fc) to the second one. As one can imagine, this notation can be generalized toany BOC or arbitrary BCS signal. It is important to note that in this particular case Tsc isequal to Tc n and f sc = nf c consequently. In addition, it can be shown that nf c = 2 f s for themodulating term as it was also the case for the usual BOC modulation. In conclusion, the PSDof MSK-BOC(fs , fc) or MSK(fs , fc) for short, will adopt the following form:GMSK-BOC( f s , fc ) ( f ) = GMSK pulse ( f )GMod BOC( f s , f c ) ( f )(4.89)where in this particular case:GMSK pulse ( f ) =8fπ3sc2(⎛ πf ⎞⎛ πf ⎞⎟⎟⎟⎟cos 2 ⎜⎜cos 2 ⎜⎜32ff4f⎝ sc ⎠ = c⎝ c⎠22222 2πfc − f 2f sc − 4 f)()(4.90)since f sc = 2 f c for the sine-phased MSK-BOC case.
On the other hand, for the even case, themodulation spectral term adopts the following form in general:⎛ πf ⎞⎛ πf ⎞sin 2 ⎜⎜ ⎟⎟sin 2 ⎜⎜ ⎟⎟⎝ fc ⎠⎝ fc ⎠ =BOCsin ( f s , f c )(f )=GMod,e⎛ πf ⎞⎛ πf ⎞⎟⎟⎟⎟ cos 2 ⎜⎜cos 2 ⎜⎜⎝ 2 fs ⎠⎝ nf c ⎠(4.91)where n is again the number of subchips in one chip. Thus, multiplying now both terms andnormalizing the power to integrate to 1 in an infinite bandwidth yields thus the PSD of theMSK(fs , fc):106GNSS Signal Structure⎛ πf ⎞⎛ πf ⎞⎟⎟ sin 2 ⎜⎜ ⎟⎟cos ⎜⎜34 fc⎝ 2 fc ⎠⎝ fc ⎠ = 1G MSK ( f s ,f c ) ( f ) =22f sc π f c2 − f 2⎛ πf ⎞ 2 f c⎟⎟cos 2 ⎜⎜⎝ 2 fs ⎠2()⎡⎛ πf ⎞⎛ πf ⎞ ⎤⎟⎟ sin ⎜⎜ ⎟⎟ ⎥⎢ 2 cos⎜⎜⎝ 2 fc ⎠⎝ fc ⎠ ⎥⎢2 fc22⎢ π fc − f⎛ πf ⎞ ⎥⎟⎟ ⎥⎢cos⎜⎜⎝ 2 f s ⎠ ⎦⎥⎣⎢(2)(4.92)For comparison the PSDs of MSK(1,1), MSK(1), BPSK(1) and BOC(1,1) are shown next:Figure 4.22. PSDs of MSK(1,1), MSK(1), BOC(1,1) and BPSK(1)As we can recognize from the previous figure, for the same chip rate of 1.023 Mcps, MSK(1)has a main lobe that is 1.5 times wider than that of BPSK(1).
On the other hand, MSK(1,1)has a main lobe that is as broad as that of BOC(1,1) but with secondary lobes that are half thewidth. MSK has a very good spectral confinement and provides at the same time constantenvelope.4.4.3Gaussian Minimum Shift Keying (GMSK)In the previous chapter the analytical expression of the MSK modulation was shown to be aparticular case of MCS with frequency modulation and sinusoidal pulse.
We have also seenthat MSK presents a very good spectral confinement in the band of interest but still animportant amount of power is allocated on the side lobes of the signal.This chapter presents a modified version of the previous MSK where the phase is furtherfiltered through a Gaussian filter to smooth the transitions from one point to the next in theconstellation. Next figure presents the GMSK generation scheme:Figure 4.23. GMSK generation scheme107GNSS Signal Structurewhere the Gaussian filter g (t ) adopts the following form in the time domain:−2πg (t ) = λBeln 22π 2 B 2 t 2ln 2(4.93)where λ is a normalization constant to maintain the power and the product BTc is the -3 dBbandwidth-symbol time product. The higher this value, the cleaner will be the eye diagram ofthe signal but more power will be transmitted on the side lobes of the spectrum.
A typicalvalue in communication applications is BTc = 0.3 which is a good compromise betweenspectral efficiency and Inter-Symbol interference.4.5Generalized(GMCS)MultilevelCodedSymbolsAs we saw in the definition of chapter 4.2, Multilevel Coded Symbols consist of spreadingwaveforms that are divided into an integer number of equal-length segments, each of themwith a deterministic value. According to this definition, any imaginable signal can in principlebe described as an MCS signal as long as the length of the segments can be expressed as arational number.
In that case, we will always be able to find a finite length n to build the MCSvector and define each of the segments with the notation used above. As we saw in previouschapter, also non-rectangular forms are possible to modulate the subchips, as long as they aredeterministic. In general, the basic pulse of each of the segments can adopt any arbitraryshape as equation (4.28) reflected.Let us now extend our definition of MCS to signal waveforms with divisions that are notnecessarily rational and thus with non finite n.
As we will see in the following chapters, thiswill open new possibilities and further simplify the notation. To do so, it is first necessary todefine an intermediate function that we will call Tertiary Coded Symbols or TCS for short.The reason for that is that many of the Generalized MCS signals can be expressed as sum ofdifferent TCS. A modified version of the TCS waveform, namely the Unilateral TCS (UTCS),is analyzed in detail in Appendix E.We further define MCS in a general case as follows:[s = s1ρ1 , s2ρ 2 , s3ρ 3 ...snρ n](4.94)where ρi indicates the part of the chip that the symbol occupies such thatn∑ρi =1i=1(4.95)This can also be seen in the following figure:108GNSS Signal StructureFigure 4.24. Generalized Multilevel Coded Symbols (GMCS)We can clearly see that in this new definition the duration of every subchip does notnecessarily have to be expressed as Tc/n but can adopt any imaginable length. Moreover, theamplitude of that subchip can adopt any amplitude too.
Finally, it is important to note that ifthe MCS generation vector [s] could have an infinite length, the SOC modulation that we sawin the previous chapter could also be considered as a particular case of the generalized MCS(GMCS) signal. In general we can also say that MCS is a particular case of GMCS withsegments of equal length.4.5.1 Tertiary Coded Symbols (TCS)Tertiary Coded Symbols are a particular case of GMCS with three-state amplitude adoptingthe values {-1,0,1}.
As shown in Appendix D, the power spectral density of a generic TCSsignal is shown to be:⎤⎡ πf(1 − ρ )⎥ nsin 2 ⎢n −1 n⎛ f ⎞ωT ⎤ ⎫⎡⎦ ⎧ s2 + 2⎣ nf cGTCS([ s ], f c ) ( f ) = ⎜⎜ c ⎟⎟sl sm cos ⎢(m − l ) c ⎥ ⎬⎨∑∑∑l2n ⎦⎭(πf )⎣l =1 m =l +1⎩ l =1⎝1− ρ ⎠(4.96)where ρ indicates the dwell time where the function adopts a value 0.
As we can see, theexpression above is practically identical to (4.23) as derived in chapter 4.2.1 except for thefactor (1-ρ). This will help us in finding explicit expressions for particular TCS modulationsin the next chapters. Moreover, it must be noted that the definition above is not only valid forbinary signals but could be extended to any amplitude in general4.5.2 Tertiary Offset Carrier (TOC)A particular case of the TCS signal is the Tertiary Offset Carrier modulation, or TOC forshort. As shown in [A.R. Pratt and J.I.R.
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